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2016-06-21 11:24 |
Speckle Phenomena in Optics: Theory and Applications
Speckle Phenomena in Optics: Theory and Applications i
LK8Wnrq "p2 $R*ie Joseph W. Goodman /Eh\07p H&3VPag Contents y%}Po)X]f 1 Origins and Manifestations of Speckle 1 1Qhx$If~ 1.1 General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 :]=Y1*L\) 1.2 Intuitive Explanation of the Cause of Speckle . . . . . . . . . . . . . . . . . . . . . . . . . 2 8X5;)h 1.3 Some Mathematical Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 |3Oe2qb 2 Random Phasor Sums 7 {5^'u^E 2.1 First and Second Moments of the Real and Imaginary Parts of the Resultant Phasor . . . . . 8 \QHe 0?6 2.2 Random Walk with a Large Number of Independent Steps . . . . . . . . . . . . . . . . . . 9 cJj0`@0f 2.3 Random Phasor Sum Plus a Known Phasor . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Ai(M06P:h 2.4 Sums of Random Phasor Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 ZRD* ^9) 2.5 Random Phasor Sums with a Finite Number of Equal-Length Components . . . . . . . . . 16 h_*=_ 2|} 2.6 Random Phasor Sums with a Nonuniform Distribution of Phases . . . . . . . . . . . . . . . 17 <MA!?7Z| 3 First-Order Statistical Properties of Optical Speckle 23 `Ft`8=( 3.1 Definition of Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 L>xcgV7 3.2 First-Order Statistics of the Intensity and Phase . . . . . . . . . . . . . . . . . . . . . . . . 24 L1P]T4a@) 3.2.1 Large Number of Random Phasors . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 .y+>-[j?B 3.2.2 Constant Phasor plus a Random Phasor Sum . . . . . . . . . . . . . . . . . . . . . 27 B~u{LvTE 3.2.3 Finite Number of Equal-Length Phasors . . . . . . . . . . . . . . . . . . . . . . . . 31 1fViW^l_ 3.3 Sums of Speckle Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 7ABHgw~?8r 3.3.1 Sums on an Amplitude Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 6(DK\58 3.3.2 Sum of Two Independent Speckle Intensities . . . . . . . . . . . . . . . . . . . . . 34 UFouIS#L 3.3.3 Sum of N Independent Speckle Intensities . . . . . . . . . . . . . . . . . . . . . . 37 0wAZ9AxA{ 3.3.4 Sums of Correlated Speckle Intensities . . . . . . . . . . . . . . . . . . . . . . . . 40 mK@\6GOMYP 3.4 Partially Polarized Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 &=X1kQG 3.5 Partially Developed Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3(E"$Se,f 3.6 Speckled Speckle, or Compound Speckle Statistics . . . . . . . . . . . . . . . . . . . . . . 47 Ny\iRU)fN 3.6.1 Speckle Driven by a Negative Exponential Intensity Distribution . . . . . . . . . . . 48 UyD=x(li 3.6.2 Speckle Driven by a Gamma Intensity Distribution . . . . . . . . . . . . . . . . . . 50 C&zgt
:q6} 3.6.3 Sums of Independent Speckle Patterns Driven by a Gamma Intensity Distribution . . 51 htMsS4^Kvd 4 Higher-Order Statistical Properties of Optical Speckle 55 08yTTt76t 4.1 Multivariate Gaussian Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 C.%iQx`
4.2 Application to Speckle Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 @UkcvhH 4.3 Multidimensional Statistics of Speckle Amplitude, Phase and Intensity . . . . . . . . . . . . 58 scJ`oc:<J 4.3.1 Joint Density Function of the Amplitudes . . . . . . . . . . . . . . . . . . . . . . . 59 E
I)Pfx"0 4.3.2 Joint Density Function of the Phases . . . . . . . . . . . . . . . . . . . . . . . . . . 60 ]/JE# 4.3.3 Joint Density Function of the Intensities . . . . . . . . . . . . . . . . . . . . . . . . 64 f!xIMIl)+ 4.4 Autocorrelation Function and Power Spectrum of Speckle . . . . . . . . . . . . . . . . . . . 66 K
,f 1c} 4.4.1 Free-Space Propagation Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ]bhzB 4.4.2 Imaging Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 [~f%z(vI 4.4.3 Speckle Size in Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 @ DU]XKv 4.5 Dependence of Speckle on Scatterer Microstructure . . . . . . . . . . . . . . . . . . . . . . 77 3ZC to[Y 4.5.1 Surface vs. Volume Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Fr/8q:m& 4.5.2 Effect of a Finite Correlation Area of the ScatteredWave . . . . . . . . . . . . . . . 78 azF"tke 4.5.3 A Regime where Speckle Size Is Independent of Scattering Spot Size . . . . . . . . 81 HEht^/pJ 4.5.4 Relation between the Correlation Areas of the ScatteredWave and the Surface Height ?n? Ep [D Fluctuations— Surface Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . 83 8^c|9ow 4.5.5 Dependence of Speckle Contrast on Surface Roughness— Surface Scattering . . . . 88 O0WzDD 4.5.6 Properties of Speckle Resulting from Volume Scattering . . . . . . . . . . . . . . . 92 e9k}n\t3 4.6 Statistics of Integrated and Blurred Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . 94 |~8iNcIS 4.6.1 Mean and Variance of Integrated Speckle . . . . . . . . . . . . . . . . . . . . . . . 95 `r+e!o 4.6.2 Approximate Result for the Probability Density Function of 9i,QCA Integrated Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 sJt&`k Z 4.6.3 “Exact” Result for the Probability Density Function of Integrated Intensity . . . . . 101 ~"
}t8`vP1 4.6.4 Integration of Partially Polarized Speckle Patterns . . . . . . . . . . . . . . . . . . . 106 H.O7Y 4.7 Statistics of Derivatives of Speckle Intensity and Phase . . . . . . . . . . . . . . . . . . . . 108 _BHb0zeot 4.7.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 "MZVwl "E# 4.7.2 Parameters for Various Scattering Spot Shapes . . . . . . . . . . . . . . . . . . . . 110 W*`2lf 4.7.3 Derivatives of Speckle Phase: Ray Directions in a Speckle Pattern . . . . . . . . . . 111 n#,AZ& 4.7.4 Derivatives of Speckle Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 :*A6Ba 4.7.5 Level Crossings of Speckle Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 116
p,]Hs{R 4.8 Zeros of Speckle Patterns: Optical Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . 118 [AE]0cO@ 4.8.1 Conditions Required for a Zero of Intensity to Occur . . . . . . . . . . . . . . . . . 119 w/h?, L| 4.8.2 Properties of Speckle Phase in the Vicinity of a Zero of Intensity . . . . . . . . . . . 119 i?4vdL8M 4.8.3 The Density of Vortices in Fully Developed Speckle . . . . . . . . . . . . . . . . . 119 t#6gjfIi 4.8.4 The Density of Vortices for Fully Developed Speckle Plus a Coherent Background . 123 k+JDbJ@ 5 OpticalMethods for Suppressing Speckle 125 4q~+K'Z 5.1 Polarization Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 fCO!M1 t 5.2 Temporal Averaging with a Moving Diffuser . . . . . . . . . . . . . . . . . . . . . . . . . 127 \W??`?Idh 5.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 niqi DT/ 5.2.2 Smooth Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 HQQc<7c", 5.2.3 Rough Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 %$!}MxUM 5.3 Wavelength and Angle Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 kTc'k 5.3.1 Free-Space Propagation, Reflection Geometry . . . . . . . . . . . . . . . . . . . . 136 bYgrKz@uK 5.3.2 Free-Space Propagation, Transmission Geometry . . . . . . . . . . . . . . . . . . . 144 p&:RSO 5.3.3 Imaging Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 L$i&>cF\_> 5.4 Temporal and Spatial Coherence Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 150 w<-CKM3qe 5.4.1 Coherence Concepts in Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 /3'-+bp^= 5.4.2 Moving Diffusers and Coherence Reduction . . . . . . . . . . . . . . . . . . . . . . 152 H.|FEV@ 5.4.3 Speckle Suppression by Reduction of Temporal Coherence . . . . . . . . . . . . . . 154 wEQV"I 5.4.4 Speckle Suppression by Reduction of Spatial Coherence . . . . . . . . . . . . . . . 157 _4x X}Z; 5.5 Use of Temporal Coherence to Destroy Spatial Coherence . . . . . . . . . . . . . . . . . . 163 J@p[v3W 5.6 Compounding Speckle Suppression Techniques . . . . . . . . . . . . . . . . . . . . . . . . 163 iNd8M V 6 Speckle in Certain Imaging Modalities 165 Tj<W4+p{ 6.1 Speckle in the Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 `<b 3e(A 6.2 Speckle in Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 M:Xswwq 6.2.1 Principles of Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 |a3)U%rUEQ 6.2.2 Speckle Suppression in Holographic Images . . . . . . . . . . . . . . . . . . . . . . 170 pWw aN4 6.3 Speckle in Optical Coherence Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . 171 D!TS/J1S;u 6.3.1 Overview of the OCT Imaging Technique . . . . . . . . . . . . . . . . . . . . . . . 172 ]\sBl 6.3.2 Analysis of OCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 oizD:| 6.3.3 Speckle and Speckle Suppression in OCT . . . . . . . . . . . . . . . . . . . . . . . 176 T$0//7$') 6.4 Speckle in Optical Projection Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 I/pavh 6.4.1 Anatomies of Projection Displays . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6b6}HO 6.4.2 Speckle Suppression in Projection Displays . . . . . . . . . . . . . . . . . . . . . . 182 "9&6bBa 6.4.3 Polarization Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6_^u}me 6.4.4 A Moving Screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 g=o)=sQd 6.4.5 Wavelength Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 \_De(
p 6.4.6 Angle Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 ~y$B#.l 6.4.7 Over-Design of the Projection Optics . . . . . . . . . . . . . . . . . . . . . . . . . 186 .v/s9'lB 6.4.8 Changing Diffuser Projected onto the Screen . . . . . . . . . . . . . . . . . . . . . 188 ~Pv4X2MO 6.4.9 Specially Designed Screens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 O}Fp\" 6.5 Speckle in Projection Microlithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 8QXxRD;0: 6.5.1 Coherence Properties of Excimer Lasers . . . . . . . . . . . . . . . . . . . . . . . 200 k
.KN9=o 6.5.2 Temporal Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 $X
WJxQRUv 6.5.3 From Exposure Fluctuations to Line Position Fluctuations . . . . . . . . . . . . . . 202 ]y2(ZTNTs 7 Speckle in Certain Non-imagingModalities 205 ;ZFn~!V 7.1 Speckle in Multimode Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Gv?'R0s 7.1.1 Modal Noise in Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 mxGa\{D#y 7.1.2 Statistics of Constrained Speckle . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 _F;(#D 7.1.3 Frequency Dependence of Modal Noise . . . . . . . . . . . . . . . . . . . . . . . . 211 q,PB;TT 7.2 Effects of Speckle on Optical Radar Performance . . . . . . . . . . . . . . . . . . . . . . . 216 do+HPnfDzU 7.2.1 Spatial Correlation of the Speckle Returned from Distant Targets . . . . . . . . . . . 217 m%qah>11 7.2.2 Speckle at Low Light Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Q<;f-9q@ 7.2.3 Detection Statistics—Direct Detection . . . . . . . . . . . . . . . . . . . . . . . . 222 Y \& 4`v' 7.2.4 Detection Statistics— Heterodyne Detection . . . . . . . . . . . . . . . . . . . . . 227 &
WYIfx{ 7.2.5 Comparison of Direct Detection and Heterodyne Detection . . . . . . . . . . . . . . 234 2%rAf8= 7.2.6 Reduction of the Effects of Speckle in Optical Radar Detection . . . . . . . . . . . . 235 6wqq"6w 7.3 Speckle and Metrology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 q/XZb@rt 8 Speckle in Imaging Through the Atmosphere 239 NyeGa 8.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 : \:jIP 8.1.1 Refractive Index Fluctuations in the Atmosphere . . . . . . . . . . . . . . . . . . . 239 600-e;p 8.2 Short-Exposure and Long-Exposure Point-Spread Functions . . . . . . . . . . . . . . . . . 240 (TVzYm
y 8.3 Long-Exposure and Short-Exposure Average Optical Transfer Functions . . . . . . . . . . . 242 ppM d 8.4 Statistical Properties of the Short-Exposure OTF and MTF . . . . . . . . . . . . . . . . . . 243 oz(<e 8.5 Astronomical Speckle Interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 wHx1CXC 8.5.1 Object Information that Is Retrievable . . . . . . . . . . . . . . . . . . . . . . . . . 248 *h-_
8.5.2 Results of a More Complete Analysis of the Form of the Speckle Transfer Function . 250 u;F++$= 8.6 The Cross-Spectrum or Knox–Thompson Technique . . . . . . . . . . . . . . . . . . . . . 252 '2WYbcU 8.6.1 The Cross-Spectrum Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . 253 hE/gul?|_ 8.6.2 Recovering Full Object Information from the Cross-Spectrum . . . . . . . . . . . . 254 s~Ni\SF 8.7 The Bispectrum Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 oi::/W|A+ 8.7.1 The Bispectrum Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 ZgcA[P 8.7.2 Recovering Full Object Information from the Bispectrum . . . . . . . . . . . . . . . 258 ]$EKowi 8.8 Speckle Correlography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 HZ>Xm6DnC5 A Linear Transformations 263 K9mL1 [B B Contrast of Partially Developed Speckle 267 I'`Q_5s5 C Statistics of Derivatives of Speckle 271 CGbW]D$@ C.1 The Correlation Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 53=VIN] C.2 Joint Density Function of the Derivatives of Phase . . . . . . . . . . . . . . . . . . . . . . . 274 0N;Pb(%7UU C.3 Joint Density Function of the Derivatives of Intensity . . . . . . . . . . . . . . . . . . . . . 274 `2o/W]SSk D Wavelength and Angle Dependence 277 |FS79Bv D.1 Free-Space Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 (fUXJ$ D.2 Imaging Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 }e9E+2}Z\ E Speckle Contrast when a Dynamic Diffuser is Projected onto a Random Screen 285 S_y!4;]ox E.1 Random Phase Diffusers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 erdWGUfQOe E.2 Diffuser that Just Fills the Projection Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 287 HfFP4#C, E.3 Diffuser Overfills the Projection Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 Gm}ecW F Statistics of Constrained Speckle 289 smoz5~ Bibliography 291 6w0/;8(_m Index 299
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